Bulletin
de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE
ON THE STRUCTURE
OF TRIANGULATED CATEGORIES
Claire Amiot
Tome 135
Fascicule 3
2007
SOCIÉTÉ MATHÉMATIQUE DE FRANCE
Publié avec le concours du Centre national de la recherche scientifique
pages 435-474
Bull. Soc. math. France
135 (3), 2007, p. 435–474
ON THE STRUCTURE OF TRIANGULATED CATEGORIES
WITH FINITELY MANY INDECOMPOSABLES
by Claire Amiot
Abstract. — We study the problem of classifying triangulated categories with finitedimensional morphism spaces and finitely many indecomposables over an algebraically
closed field k. We obtain a new proof of the following result due to Xiao and Zhu:
the Auslander-Reiten quiver of such a category T is of the form Z∆/G where ∆ is
a disjoint union of simply-laced Dynkin diagrams and G a weakly admissible group
of automorphisms of Z∆. Then we prove that for ‘most’ groups G, the category T
is standard, i.e. k-linearly equivalent to an orbit category Db ( mod k∆)/Φ. This
happens in particular when T is maximal d-Calabi-Yau with d ≥ 2. Moreover, if T
is standard and algebraic, we can even construct a triangle equivalence between T
and the corresponding orbit category. Finally we give a sufficient condition for the
category of projectives of a Frobenius category to be triangulated. This allows us to
construct non standard 1-Calabi-Yau categories using deformed preprojective algebras
of generalized Dynkin type.
Texte reçu le 16 janvier 2007, accepté le 1er juin 2007
Claire Amiot, Université Paris 7, Institut de Mathématiques de Jussieu, Théorie des
groupes et des représentations, Case 7012, 2 Place Jussieu, 75251 Paris Cedex 05 (France)
E-mail : [email protected]
2000 Mathematics Subject Classification. — 18E30, 16G70.
Key words and phrases. — Locally finite triangulated category, Calabi-Yau category, Dynkin
diagram, Auslander-Reiten quiver, orbit category.
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Résumé (Sur la structure des catégories triangulées). — Cet article traite du problème de classification des catégories triangulées sur un corps algébriquement clos k
dont les espaces de morphismes sont de dimension finie et avec un nombre fini d’indécomposables. Nous obtenons une nouvelle preuve du résultat suivant dû à Xiao et
Zhu : le carquois d’Auslander-Reiten d’une telle catégorie T est de la forme Z∆/G
où ∆ est une union disjointe de diagrammes de Dynkin simplement lacés et G est un
groupe d’automorphismes de Z∆ faiblement admissible. Nous montrons ensuite que
pour ‘presque’ tous groupes G, la catégorie T est standard, c’est-à-dire k-linéairement
équivalente à une catégorie d’orbites Db (mod k∆)/Φ. C’est en particulier le cas lorsque
T est maximale d-Calabi-Yau avec d ≥ 2. De plus, si T est standard et algébrique,
nous pouvons même construire une équivalence triangulée entre T et la catégorie d’orbites correspondante. Nous donnons finalemant une condition suffisante pour que la
catégorie de projectifs d’une catégorie de Frobenius soit triangulée. Cela nous permet de construire des catégories 1-Calabi-Yau non standard en utilisant les algèbres
préprojectives déformées de type Dynkin généralisé.
Introduction
Let k be an algebraically closed field and T a small Krull-Remak-Schmidt
k-linear triangulated category (see [47]). We assume that
a) T is Hom-finite, i.e. the space HomT (X, Y ) is finite-dimensional for all
objects X, Y of T .
It follows that indecomposable objects of T have local endomorphism rings
and that each object of T decomposes into a finite direct sum of indecomposables [17, 3.3]. We assume moreover that
b) T is locally finite, i.e. for each indecomposable X of T , there are at most
finitely many isoclasses of indecomposables Y such that HomT (X, Y ) 6= 0.
It was shown in [48] that condition b) implies its dual. Condition b) holds
in particular if we have
b0 ) T is additively finite, i.e. there are only finitely many isomorphism classes
of indecomposables in T .
The study of particular classes of such triangulated categories T has a long
history. Let us briefly recall some of its highlights:
1) If A is a representation-finite selfinjective algebra, then the stable category T of finite-dimensional (right) A-modules satisfies our assumptions and
is additively finite. The structure of the underlying k-linear category of T was
determined by C. Riedtmann in [39], [40], [41] and [42].
2) In [21], D. Happel showed that the bounded derived category of the category of finite-dimensional representations of a representation-finite quiver is
locally finite and described its underlying k-linear category.
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3) The stable category CM(R) of Cohen-Macaulay modules over a commutative complete local Gorenstein isolated singularity R of dimension d is a
Hom-finite triangulated category which is (d − 1)-Calabi-Yau (cf. for example
[28] and [50]). In [4], M. Auslander and I. Reiten showed that if the dimension
of R is 1, then the category CM(R) is additively finite and computed the shape
of the components of its Auslander-Reiten quiver.
4) The cluster category CQ of a finite quiver Q without oriented cycles was
introduced in [12] if Q is an orientation of a Dynkin diagram of type A and
in [11] in the general case. The category CQ is triangulated [30] and, if Q is
representation-finite, satisfies a) and b0 ).
In a recent article [48], J. Xiao and B. Zhu determined the structure of the
Auslander-Reiten quiver of a locally finite triangulated category. In this paper,
we obtain the same result with a new proof in Section 4, namely that each
connected component of the Auslander-Reiten quiver of the category T is of
the form Z∆/G, where ∆ is a simply-laced Dynkin diagram and G is trivial or
a weakly admissible group of automorphisms. Contrary to J. Xiao and B. Zhu,
we do not discuss separately the case where the Auslander-Reiten contains
a loop.
We are interested in the k-linear structure of T . If the Auslander-Reiten
quiver of T is of the form Z∆, we show that the category T is standard, i.e. it
is equivalent to the mesh category k(Z∆). Then in Section 6, we prove that T
is standard if the number of vertices of Γ = Z∆/G is strictly greater than
the number of isoclasses of indecomposables of mod k∆. In the last section,
using [8] we construct examples of non standard triangulated categories such
that Γ = Z∆/τ .
Finally, in the standard cases, we are interested in the triangulated structure
of T . For this, we need to make additional assumptions on T . If the AuslanderReiten quiver is of the form Z∆, and if T is the base of a tower of triangulated
categories [29], we show that there is a triangle equivalence between T and the
derived category Db (mod k∆). For the additively finite cases, we have to assume that T is standard and algebraic in the sense of [31]. We then show that T
is (algebraically) triangle equivalent to the orbit category of Db (mod k∆) under
the action of a weakly admissible group of automorphisms. In particular, for
each d ≥ 2, the algebraic triangulated categories with finitely many indecomposables which are maximal Calabi-Yau of CY-dimension d are parametrized
by the simply-laced Dynkin diagrams.
Our results apply in particular to many stable categories mod A of
representation-finite selfinjective algebras A. These algebras were classified up to stable equivalence by C. Riedtmann [40], [42] and H. Asashiba [1].
In [9], J. Białkowski and A. Skowroński give a necessary and sufficient condition
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on these algebras so that their stable categories mod A are Calabi-Yau. In [26]
and [27], T. Holm and P. Jørgensen prove that certain stable categories mod A
are in fact d-cluster categories. These results can also be proved using our
Corollary 7.3.
This paper is organized as follows: In Section 1, we prove that T has
Auslander-Reiten triangles. Section 2 is dedicated to definitions about stable valued translation quivers and admissible automorphisms groups [23], [24],
[14]. We show in Section 3 that the Auslander-Reiten quiver of T is a stable
valued quiver and in Section 4, we reprove the result of J. Xiao and B. Zhu [48]:
The Auslander-Reiten quiver is a disjoint union of quivers Z∆/G, where ∆ is
a Dynkin quiver of type A, D or E, and G a weakly admissible group of automorphisms. In Section 5, we construct a covering functor Db (mod k∆) → T
using Riedtmann’s method [39]. Then, in Section 6, we exhibit some combinatorial cases in which T has to be standard, in particular when T is maximal
d-Calabi-Yau with d ≥ 2. Section 7 is dedicated to the algebraic case. If T
is algebraic and standard, we can construct a triangle equivalence between T
and an orbit category. If P is a k-category such that mod P is a Frobenius
category satisfying certain conditions, we will prove in Section 8 that P has
naturally a triangulated structure. This allows us to deduce in Section 9 that
the category proj P f (∆) of the projective modules over a deformed preprojective algebra of generalized Dynkin type [8] is naturally triangulated and to
reduce the classification of the additively finite triangulated categories which
are 1-Calabi-Yau to that of the deformed preprojective algebras in the sense
of [8]. In particular, thanks to [8], we obtain the existence of non standard 1Calabi-Yau categories in characteritic 2. Using our results and an extension of
those of [8], Białkowski and Skowroński have recently proved [10] the existence
of non standard 1-Calabi-Yau categories in characteristic 3. This is noteworthy
since in characteristic different from 2, additively finite module categories are
always standard [6].
Acknowledgments. — I would like to thank my supervisor B. Keller for his availability and many helpful discussions, and P. Jørgensen for his interest in this
work and for suggesting some clarifications to me in the proof of Theorem 7.2.
Notation and terminology
We work over an algebraically closed field k. By a triangulated category,
we mean a k-linear triangulated category T . We write S for the suspension
u
v
w
functor of T and U −
→ V −
→ W −
→ SU for a distinguished triangle. We
say that T is Hom-finite if for each pair X, Y of objects in T , the space
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HomT (X, Y ) is finite-dimensional over k. The category T will be called a KrullRemak-Schmidt category if each object is isomorphic to a finite direct sum of
indecomposable objects with unicity (up to reordering) of this decomposition,
and if the endomorphism ring of an indecomposable object is a local ring. This
implies that idempotents of T split, i.e. if e is an idempotent of X, then
e = σρ where σ is a section and ρ is a retraction [22, I, 3.2]. The category T
will be called locally finite if for each indecomposable X of T , there are only
finitely many isoclasses of indecomposables Y such that HomT (X, Y ) 6= 0. This
property is selfdual by [48, prop. 1.1].
The Serre functor will be denoted by ν (see definition in Section 1). The
Auslander-Reiten translation will always be denoted by τ (Section 1).
Let T and T 0 be two triangulated categories. An S-functor (F, φ) is given
by a k-linear functor F : T → T 0 and a functor isomorphism φ between the
functors F ◦ S and S 0 ◦ F , where S is the suspension of T and S 0 the suspension
of T 0 . The notion of ν-functor, or τ -functor is then clear. A triangle functor is
u
v
w
an S-functor (F, φ) such that for each triangle U −
→V −
→W −
→ SU of T , the
Fu
Fv
φU ◦F w
sequence F U −−→ F V −−→ F W −−−−−→ S 0 F U is a triangle of T 0 .
The category T is Calabi-Yau if there exists an integer d > 0 such that we
have a triangle functor isomorphism between S d and ν. We say that T is maximal d-Calabi-Yau if T is d-Calabi-Yau and if for each covering functor T 0 → T
with T 0 d-Calabi-Yau, we have a k-linear equivalence between T and T 0 .
For an additive k-category E, we write mod E for the category of contravariant finitely presented functors from E to mod k (Section 8), and if the projectives of mod E coincide with the injectives, mod E will be the stable category.
1. Serre duality and Auslander-Reiten triangles
1.1. Serre duality. — Recall from [38] that a Serre functor for T is an autoequivalence ν : T → T together with an isomorphism D HomT (X, ?) '
HomT (?, νX) for each X ∈ T , where D is the duality Homk (?, k).
Theorem 1.1. — Let T be a Krull-Remak-Schmidt, locally finite triangulated
category. Then T has a Serre functor ν.
Proof. — Let X be an object of T . We write X ∧ for the functor HomT (?, X)
and F for the functor D HomT (X, ?). Using the lemma [38, I.1.6] we just have
to show that F is representable. Indeed, the category T op is locally finite as
well. The proof is in two steps.
Step 1: The functor F is finitely presented. — Let Y1 , . . . , Yr be representatives of the isoclasses of indecomposable objects of T such that F Yi is not zero.
The space Hom(Yi∧ , F ) is finite-dimensional over k. Indeed it is isomorphic
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to F Yi by the Yoneda lemma. Therefore, the functor Hom(Yi∧ , F ) ⊗k Yi∧ is
representable. We get an epimorphism from a representable functor to F :
r
M
Hom(Yi∧ , F ) ⊗k Yi∧ −→ F.
i=1
By applying the same argument to its kernel we get a projective presentation
of F of the form U ∧ → V ∧ → F → 0, with U and V in T .
Step 2: A cohomological functor H : T op → mod k is representable if and
only if it is finitely presented. — Let
u∧
φ
U ∧ −−→ V ∧ −
→H→0
u
v
w
be a presentation of H. We form a triangle U −
→V −
→W −
→ SU. We get an
exact sequence
u∧
v∧
w∧
U ∧ −−→ V ∧ −−→ W ∧ −−→ (SU )∧ .
Since the composition of φ with u∧ is zero and H is cohomological, the morphism φ factors through v ∧ . But H is the cokernel of u∧ , so v ∧ factors
through φ. We obtain a commutative diagram
The equality φ0 ◦ i ◦ φ = φ0 ◦ v ∧ = φ implies that φ0 ◦ i is the identity of H
because φ is an epimorphism. We deduce that H is a direct factor of W ∧ . The
composition i ◦ φ0 = e∧ is an idempotent. Then e ∈ End(W ) splits and we
get H = W 0∧ for a direct factor W 0 of W .
1.2. Auslander-Reiten triangles
u
v
w
Definition 1.2.1 (see [21]). — A triangle X −
→Y −
→Z−
→ SX of T is called
an Auslander-Reiten triangle or AR-triangle if the following conditions are
satisfied:
(AR1) X and Z are indecomposable objects;
(AR2) w 6= 0;
(AR3) if f : W → Z is not a retraction, there exists f 0 : W → Y such that
vf 0 = f ;
(AR30 ) if g : X → V is not a section, there exists g 0 : Y → V such that
g 0 u = g.
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Let us recall that, if (AR1) and (AR2) hold, the conditions (AR3) and (AR30 )
are equivalent. We say that a triangulated category T has Auslander-Reiten
triangles if, for any indecomposable object Z of T , there exists an AR-triangle
u
v
w
ending at Z: X −
→Y −
→Z−
→ SX. In this case, the AR-triangle is unique up
to triangle isomorphism inducing the identity of Z.
The following proposition is proved in [38, Prop. I.2.3].
Proposition 1.2. — Let T be a Krull-Remak-Schmidt, locally finite triangulated category. Then the category T has Auslander-Reiten triangles.
The composition τ = S −1 ν is called the Auslander-Reiten translation. An
AR-triangle of T ending at Z has the form
u
v
w
τZ −
→Y −
→Z−
→ νZ.
2. Valued translation quivers and automorphism groups
2.1. Translation quivers. — We recall some definitions and notations concerning
quivers [14]. A quiver Q = (Q0 , Q1 , s, t) is given by the set Q0 of its vertices,
the set Q1 of its arrows, a source map s and a tail map t. If x ∈ Q0 is a vertex,
we denote by x+ the set of direct successors of x, and by x− the set of its direct
predecessors. We say that Q is locally finite if for each vertex x ∈ Q0 , there are
finitely many arrows ending at x and starting at x (in this case, x+ and x− are
finite sets). The quiver Q is said to be without double arrows, if two different
arrows cannot have the same tail and source.
Definition 2.1.1. — A stable translation quiver (Q, τ ) is a locally finite
quiver without double arrows with a bijection τ : Q0 → Q0 such that
(τ x)+ = x−
for each vertex x.
For each arrow α : x → y, let σα be the unique arrow τ y → x.
Note that a stable translation quiver can have loops.
Definition 2.1.2. — A valued translation quiver (Q, τ, a) is a stable translation quiver (Q, τ ) with a map a : Q1 → N such that
a(α) = a(σα) for each arrow α.
If α is an arrow from x to y, we write axy instead of a(α).
Definition 2.1.3. — Let ∆ be an oriented tree. The repetition of ∆ is the
quiver Z∆ defined as follows:
• (Z∆)0 = Z × ∆0 ,
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• (Z∆)1 = Z × ∆1 ∪ σ(Z × ∆1 ) with arrows
(n, α) : (n, x) −→ (n, y) and σ(n, α) : (n − 1, y) → (n, x)
for each arrow α : x → y of ∆.
The quiver Z∆ with the translation τ (n, x) = (n − 1, x) is clearly a stable
translation quiver which does not depend (up to isomorphism) on the orientation of ∆ (see [39]).
2.2. Groups of weakly admissible automorphisms
Definition 2.2.1. — An automorphism group G of a quiver is said to be
admissible [39] if no orbit of G intersects a set of the form {x}∪x+ or {x}∪x− in
more than one point. It said to be weakly admissible [14] if, for each g ∈ G−{1}
and for each x ∈ Q0 , we have x+ ∩ (gx)+ = ∅.
Note that an admissible automorphism group is a weakly admissible automorphism group. Let us fix a numbering and an orientation of the simply-laced
Dynkin trees.
Let ∆ be a Dynkin tree. We define an automorphism S of Z∆ as follows:
— if ∆ = An , then S(p, q) = (p + q, n + 1 − q);
— if ∆ = Dn with n even, then S = τ −n+1 ;
— if ∆ = Dn with n odd, then S = τ −n+1 φ where φ is the automorphism
of Dn which exchanges n and n − 1;
— if ∆ = E6 , then S = φτ −6 where φ is the automorphism of E6 which
exchanges 2 and 5, and 1 and 6;
— if ∆ = E7 , then S = τ −9 ;
— and if ∆ = E8 , then S = τ −15 .
In [39, Anhang 2], Riedtmann describes all admissible automorphism groups
of Dynkin diagrams. Here is a more precise result in which we describe all
weakly admissible automorphism groups of Dynkin diagrams.
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Theorem 2.1. — Let ∆ be a Dynkin tree and G a non trivial group of weakly
admissible automorphisms of Z∆. Then G is isomorphic to Z, and here is a
list of its possible generators:
— if ∆ = An with n odd, possible generators are τ r and φτ r with r ≥ 1,
1
where φ = τ 2 (n+1) S is an automorphism of Z∆ of order 2;
— if ∆ = An with n even, then possible generators are ρr , where r ≥ 1 and
1
where ρ = τ 2 n S (since ρ2 = τ −1 , τ r is a possible generator);
— if ∆ = Dn with n ≥ 5, then possible generators are τ r and τ r φ, where
r ≥ 1 and where φ = (n − 1, n) is the automorphism of Dn exchanging n
and n − 1;
— if ∆ = D4 , then possible generators are φτ r , where r ≥ 1 and where φ
belongs to S3 the permutation group on three elements seen as subgroup
of automorphisms of D4 ;
— if ∆ = E6 , then possible generators are τ r and φτ r , where r ≥ 1 and
where φ is the automorphism of E6 exchanging 2 and 5, and 1 and 6;
— if ∆ = En with n = 7, 8, possible generators are τ r , where r ≥ 1.
The unique weakly admissible automorphism group which is not admissible
exists for An , n even, and is generated by ρ.
3. Property of the Auslander-Reiten translation
We define the Auslander-Reiten quiver ΓT of the category T as a valued
quiver (Γ, a). The vertices are the isoclasses of indecomposable objects. Given
two indecomposable objects X and Y of T , we draw one arrow from x = [X]
to y = [Y ] if the vector space R(X, Y )/R2 (X, Y ) is not zero, where R(?, ?) is
the radical of the bifunctor HomT (?, ?). A morphism of R(X, Y ) which does
not vanish in the quotient R(X, Y )/R2 (X, Y ) will be called irreducible. Then
we put
axy = dimk R(X, Y )/R2 (X, Y ).
Remark that the fact that T is locally finite implies that its AR-quiver is
locally finite. The aim of this section is to show that ΓT with the translation τ
defined in the first part is a valued translation quiver. In other words, we want
to show the proposition:
Proposition 3.1. — If X and Y are indecomposable objects of T , we have
dimk R(X, Y )/R2 (X, Y ) = dimk R(τ Y, X)/R2 (τ Y, X).
Let us recall some definitions [22].
Definition 3.0.2. — A morphism g : Y → Z is called sink morphism if the
following hold
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1) g is not a retraction;
2) if h : M → Z is not a retraction, then h factors through g;
3) if u is an endomorphism of Y which satisfies gu = g, then u is an automorphism.
Dually, a morphism f : X → Y is called source morphism if the following hold:
1) f is not a section;
2) if h : X → M is not a section, then h factors through f ;
3) if u is an endomorphism of Y which satisfies uf = f , then u is an automorphism.
These conditions imply that X and Z are indecomposable. Obviously, if
u
v
w
X −
→Y −
→Z −
→ SX is an AR-triangle, then u is a source morphism and v
is a sink morphism. Conversely, if v ∈ HomT (Y, Z) is a sink morphism (or
if u ∈ HomT (X, Y ) is a source morphism), then there exists an AR-triangle
u
v
w
X−
→Y −
→Z−
→ SX (see [22, I, 4.5]).
The following lemma (and the dual statement) is proved in [43, 2.2.5].
Lemma 3.2. — Let g be a morphism from Y to Z, where Z is indecomposable
L
and Y = ri=1 Yini is the decomposition of Y into indecomposables. Then the
morphism g is a sink morphism if and only if the following hold:
1) For each i = 1, . . . , r and j = 1, . . . , ni , the restriction gi,j of g to the j-th
component of the i-th isotopic part of Y belongs to the radical R(Yi , Z).
2) For each i = 1, . . . , r, the family (g i,j )j=1,...,ni forms a k-basis of the
space R(Yi , Z)/R2 (Yi , Z).
3) If h ∈ HomT (Y 0 , Z) is irreducible and Y 0 indecomposable, then h factors
through g and Y 0 is isomorphic to Yi for some i.
Using this lemma, it is easy to see that Proposition 3.1 holds. Thus, the
Auslander-Reiten quiver ΓT = (Γ, τ, a) of the category T is a valued translation
quiver.
4. Structure of the Auslander-Reiten quiver
This section is dedicated to another proof of a theorem due to J. Xiao and
B. Zhu:
Theorem 4.1 (see [49]). — Let T be a Krull-Remak-Schmidt, locally finite
triangulated category. Let Γ be a connected component of the AR-quiver of T .
Then there exists a Dynkin tree ∆ of type A, D or E, a weakly admissible automorphism group G of Z∆ and an isomorphism of valued translation quivers
∼
θ:Γ−
→ Z∆/G.
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The underlying graph of the tree ∆ is unique up to isomorphism (it is called
the type of Γ), and the group G is unique up to conjugacy in Aut(Z∆). In
particular, if T has an infinite number of isoclasses of indecomposable objects,
then G is trivial, and Γ is the repetition quiver Z∆.
4.1. Auslander-Reiten quivers with a loop. — In this section, we suppose that the
Auslander-Reiten quiver of T contains a loop, i.e. there exists an arrow with
same tail and source. Thus, we suppose that there exists an indecomposable
X of T such that
dimk R(X, X)/R2 (X, X) ≥ 1.
Proposition 4.2. — Let X be an indecomposable object of T . Suppose that
we have dimk R(X, X)/R2 (X, X) ≥ 1. Then τ X is isomorphic to X.
To prove this, we need a lemma.
Lemma 4.3. — Let
f1
f2
fn
X1 −→ X2 −→ · · · −→ Xn+1
be a sequence of irreducible morphisms between indecomposable objects
with n ≥ 2. If the composition fn ◦ fn−1 ◦ · · · ◦ f1 is zero, then there exists an i such that τ −1 Xi is isomorphic to Xi+2 .
Proof. — The proof proceeds by induction on n. Let us show the assertion
f1
f2
for n = 2. Suppose X1 −→ X2 −→ X3 is a sequence such that f2 ◦ f1 = 0. We
can then construct an AR-triangle:
The composition f2 ◦ f1 is zero, thus the morphism f2 factors through g1 . As
the morphisms g1 and f2 are irreducible, we conclude that β is a retraction,
and X3 a direct summand of τ −1 X1 . But X1 is indecomposable, so β is an
isomorphism between X3 and τ −1 X1 .
Now suppose that the property holds for an integer n − 1 and that we
have fn ◦ fn−1 ◦ · · · ◦ f1 = 0. If the composition fn−1 ◦ · · · ◦ f1 is zero, the
proposition holds by induction. So we can suppose that for i ≤ n − 2, the
objects τ −1 Xi and Xi+2 are not isomorphic. We show now by induction
on i that for each i ≤ n − 1, there exists a map βi : τ −1 Xi → Xn+1 such
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that fn ◦ · · · ◦ fi+1 = βi gi where gi : Xi+1 → τ −1 Xi is an irreducible morphism. For i = 1, we construct an AR-triangle:
As the composition fn ◦ · · · ◦ f1 is zero, we have the factorization fn ◦ · · · ◦ f2 =
β1 g1 .
Now for i, as τ −1 Xi−1 is not isomorphic to Xi+1 , there exists an AR-triangle
of the form
By induction, −βi−1 gi−1 + fn ◦ · · · ◦ fi+1 fi is zero, thus fn ◦ · · · ◦ fi+1 factors
through gi . This property is true for i = n − 1, so we have a map βn−1 :
τ −1 Xn−1 → Xn+1 such that βn−1 gn−1 = fn . As gn−1 and fn are irreducible,
we conclude that βn−1 is an isomorphism between Xn+1 and τ −1 Xn−1 .
Now we are able to prove Proposition 4.2. There exists an irreducible map
f : X → X. Suppose that X and τ X are not isomorphic. Then from the
previous lemma, the endomorphism f n is non zero for each n. But since T is a
Krull-Remak-Schmidt, locally finite category, a power of the radical R(X, X)
vanishes. This is a contradiction.
e = (Γ
e0 , Γ
e 1 , ã) be the valued quiver obtained
4.2. Proof of Theorem 4.1. — Let Γ
from Γ by removing the loops, i.e. we have
e 0 = Γ0 , Γ
e 1 = α ∈ Γ1 such that s(α) 6= t(α) , and ã = a Γ̃ .
Γ
1
e = (Γ
e0 , Γ
e 1 , ã) with the translation τ is a valued
Lemma 4.4. — The quiver Γ
translation quiver without loop.
Proof. — We have to check that the map σ is well-defined. But from Proposition 4.2, if α is a loop on a vertex x, σ(α) is the unique arrow from τ x = x
e is obtained from Γ by removing some σ-orbits and
to x, i.e. σ(α) = α. Thus Γ
it keeps the structure of stable valued translation quiver.
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Now, we can apply Riedtmann’s Struktursatz [39] and the result of HappelPreiser-Ringel [24]. There exist a tree ∆ and an admissible automorphism
e is isomorphic to Z∆/G as a
group G (which may be trivial) of Z∆ such that Γ
valued translation quiver. The underlying graph of the tree ∆ is then unique
up to isomorphism and the group G is unique up to conjugacy in Aut(Z∆).
Let x be a vertex of ∆. We write x for the image of x by the map:
π
e ,−→ Γ.
∆ → Z∆ −
→ Z∆/G ' Γ
Let C : ∆0 × ∆0 → Z be the matrix defined as follows:
• C(x, y) = −a x y (resp. −a y x ) if there exists an arrow from x to y (resp.
from y to x) in ∆,
• C(x, x) = 2 − a x x ,
• C(x, y) = 0 otherwise.
The matrix C is symmetric; it is a ‘generalized Cartan matrix’ in the sense
of [23]. If we remove the loops from the ‘underlying graph of C’ (in the
sense of [23]), we get the underlying graph of ∆.
In order to apply the result of Happel-Preiser-Ringel [23, Section 2], we have
to show:
Lemma 4.5. — The set ∆0 of vertices of ∆ is finite.
Proof. — Riedtmann’s construction of ∆ is the following. We fix a vertex x0
e 0 . Then the vertices of ∆ are the paths of Γ
e beginning on x0 and which
in Γ
e 1 . Now suppose
do not contain subpaths of the form ασ(α), where α is in Γ
that ∆0 is an infinite set. Then for each n, there exists a sequence
α
α
αn−1
α
1
2
n
x0 −→
x1 −→
· · · −−−→ xn−1 −−→
xn
such that τ xi+2 6= xi . Then there exist some indecomposables X0 , . . . , Xn
such that the vector space R(Xi−1 , Xi )/R2 (Xi−1 , Xi ) is not zero. Thus from
Lemma 4.3, there exists irreducible morphisms fi : Xi−1 → Xi such that the
composition fn ◦ fn−1 ◦ · · · ◦ f1 does not vanish. But the functor HomT (X0 , ?)
has finite support. Thus there is an indecomposable Y which appears an infinite
number of times in the sequence (Xi )i . But since RN (Y, Y ) vanishes for an N ,
we have a contradiction.
Let S a system of representatives of isoclasses of indecomposables of T . For
an indecomposable Y of T , we put
X
`(Y ) =
dimk HomT (M, Y ).
M ∈S
This sum is finite since T is locally finite.
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Lemma 4.6. — For x in ∆0 , we write dx = `(x). Then for each x ∈ ∆0 , we
have
X
dy Cxy = 2.
y∈∆0
Proof. — Let X and U be indecomposables of T . Let
u
v
w
X−
→Y −
→Z−
→ SX
be an AR-triangle. We write (U, ?) for the cohomological functor HomT (U, ?).
Thus, we have a long exact sequence
S −1 w
w
v
u
∗
∗
∗
∗
(U, SX).
(U, Z) −−→
(U, Y ) −→
(U, X) −→
(U, S −1 Z) −−−−→
Let SZ (U ) be the image of the map w∗ . We have the exact sequence:
u
v
w
∗
∗
∗
0 → SS −1 Z (U ) → (U, X) −→
(U, Y ) −→
(U, Z) −−→
SZ (U ) → 0.
Thus we have the equality
dimk SZ (U ) + dimk SS −1 Z (U ) + dimk (U, Y ) = dimk (U, X) + dimk (U, Z).
If U is not isomorphic to Z, each map from U to Z is radical, thus SZ (U )
is zero. If U is isomorphic to Z, the map w∗ factors through the radical of
End(Z), so SZ (Z) is isomorphic to k. Then summing the previous equality
when U runs over S, we get
`(X) + `(Z) = `(Y ) + 2.
Clearly ` is τ -invariant, thus `(Z) equals `(X). If the decomposition of Y is of
L
the form ri=1 Yini , we get
X
X
`(Y ) =
ni `(Yi ) =
aXYi `(Yi ) + aXX `(X).
i
i,X→Yi ∈Γ̃
We deduce the formula
2 = (2 − aXX )`(X) −
X
aXYi `(Yi ).
i,X→Yi ∈Γ̃
e Then an arrow x → Y in Γ
e
Let x be a vertex of the tree ∆ and x its image in Γ.
comes from an arrow (x, 0) → (y, 0) in Z∆ or from an arrow (x, 0) → (y, −1)
in Z∆, i.e. from an arrow (y, 0) → (x, 0). Indeed the projection Z∆ → Z∆/G
is a covering. From this we deduce the equality
X
X
X
2 = (2 − ax x )dx −
ax y dy −
ay x dy =
dy Cxy .
y,x→y∈∆
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Now we can prove Theorem 4.1. The matrix C is a ‘generalized Cartan
matrix’. The previous lemma gives us a subadditive function which is not
additive. Thus by [23], the underlying graph of C is of ‘generalized Dynkin
type’. As C is symmetric, the graph is necessarily of type A, D, E, or L. But
this graph is the graph ∆ with the valuation a. We are done in the cases A, D,
or E.
The case Ln occurs when the AR-quiver contains at least one loop. We can
see Ln as An with valuations on the vertices with a loop. Then, it is obvious
that the automorphism groups of ZLn are generated by τ r for an r ≥ 1. But
Proposition 4.2 tell us that a vertex x with a loop satisfies τ x = x. Thus G is
generated by τ and the AR-quiver has the following form:
This quiver is isomorphic to the quiver ZA2n /G where G is the group generated
by the automorphism τ n S = ρ.
The suspension functor S sends the indecomposables on indecomposables,
thus it can be seen as an automorphism of the AR-quiver. It is exactly the
automorphism S defined in Section 2.2.
As shown in [49], it follows from the results of [30] that for each Dynkin
tree ∆ and for each weakly admissible group of automorphisms G of Z∆, there
exists a locally finite triangulated category T such that ΓT ' Z∆/G. This
category is of the form T = Db (mod k∆)/ϕ where ϕ is an auto-equivalence
of Db (mod k∆).
5. Construction of a covering functor
From now, we suppose that the AR-quiver Γ of T is connected. We know its
structure. It is natural to ask: Is the category T standard, i.e. equivalent as
a k-linear category to the mesh category k(Γ)? First, in this part we construct a
covering functor F : k(Z∆) → T .
5.1. Construction. — We write π : Z∆ → Γ for the canonical projection. As
G is a weakly admissible group, this projection verifies the following property:
if x is a vertex of Z∆, the number of arrows of Z∆ with source x is equal
to the number of arrows of Z∆/G with source πx. Let S be a system of
representatives of the isoclasses of indecomposables of T . We write ind T for
the full subcategory of T whose set of objects is S. For a tree ∆, we write k(Z∆)
for the mesh category (see [39]). Using the same proof as Riedtmann [39], one
shows the following theorem.
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Theorem 5.1. — There exists a k-linear functor F : k(Z∆) → ind T which
is surjective and induces bijections:
M
Homk(Z∆) (x, z) −→ HomT (F x, F y),
F z=F y
for all vertices x and y of Z∆.
5.2. Infinite case. — If the category T is locally finite not finite i.e. if there
is infinitely many indecomposables, the constructed functor F is immediately
fully faithful. Thus we get the corollary.
Corollary 5.2. — If ind T is not finite, then we have a k-linear equivalence
between T and the mesh category k(Z∆).
5.3. Uniqueness criterion. — The covering functor F can be seen as a k-linear
functor from the derived category Db (mod k∆) to the category T . By construction, it satisfies the following property called the AR-property:
f
g
h
For each AR-triangle X −
→Y −
→Z −
→ SX of Db (mod k∆), there exists a
Ff
Fg
triangle of T of the form F X −−→ F Y −−→ F Z →
− SF X.
In fact, thanks to this property, F is determined by its restriction to the
subcategory proj k∆ = k(∆), i.e. we have the following lemma.
Lemma 5.3. — Let F and G be k-linear functors from Db (mod k∆) to T .
Suppose that F and G satisfy the AR-property and that the restrictions F k(∆)
and G k(∆) are isomorphic. Then the functors F and G are isomorphic as
k-linear functors.
Proof. — It is easy to construct this isomorphism by induction using the (TR3)
axiom of the triangulated categories (see [36]).
6. Particular cases of k-linear equivalence
From now we suppose that the category T is finite, i.e. T has finitely many
isoclasses of indecomposable objects.
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6.1. Equivalence criterion. — Let Γ be the AR-quiver of T and suppose that it is
isomorphic to Z∆/G. Let ϕ be a generator of G. It induces an automorphism
in the mesh category k(Z∆) that we still denote by ϕ. Then we have the
following equivalence criterion.
Proposition 6.1. — The categories k(Γ) and ind T are equivalent as kcategories if and only if there exists a covering functor F : k(Z∆) → ind T and
an isomorphism of functors Φ : F ◦ ϕ → F .
The proof consists in constructing a k-linear equivalence between ind T and
the orbit category k(Z∆)/ϕZ using the universal property of the orbit category
(see [30]), and then constructing an equivalence between k(Z∆)/ϕZ and k(Γ).
6.2. Cylindric case for An
Theorem 6.2. — If ∆ = An and ϕ = τ r for some r ≥ 1, then there exists
a functor isomorphism Φ : F ◦ ϕ → F , i.e. for each object x of k(Z∆) there
exists an automorphism Φx of F x such that for each arrow α : x → y of Z∆,
the following diagram commutes:
Φ
x
F x −−−−
→


Fα y
Fx


y F ϕα
Φy
F y −−−−→ F y.
To prove this, we need the following lemma.
Lemma 6.3. — Let α : x → y be an arrow of ZAn and let c be a path from x
to τ r y, r ∈ Z, which is not zero in the mesh category k(ZAn ). Then c can be
written c0 α where c0 is a path from y to τ r y (up to sign).
Proof of the lemma. — There is a path from x to τ r y, thus, we have
Homk(Z∆) (x, τ r y) ' k, and x and τ r y are opposite vertices of a ‘rectangle’ in ZAn . This implies that there exists a path from x to τ r y beginning
by α.
Proof of Theorem 6.2. — Combining Proposition 6.1 and Lemma 5.3, we have
just to construct an isomorphism between the restriction of F and F ◦ ϕ to a
subquiver An .
Let us fix a full subquiver of ZAn of the following form
α
α
αn−1
1
2
x1 −→
x2 −→
· · · −−−→ xn
such that x1 , . . . , xn are representatives of the τ -orbits in ZAn . We define
the (Φxi )i=1...n by induction. We fix Φx1 = IdF x1 . Now suppose we have
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constructed some automorphisms Φx1 , . . . , Φxi such that for each j ≤ i the
following diagram is commutative:
Φxj−1
F xj−1 −−−−→ F xj−1




F αj−1 y
y F ϕαj−1
Φxj
F xj −−−−−→ F xj .
The composition (F ϕαi ) ◦ Φxi is in the morphism space HomT (F xi , F xi+1 ),
which is isomorphic, by Theorem 5.1, to the space
M
Homk(Z∆) (xi , z).
F z=F xi+1
Thus we can write
(F ϕαi )Φxi = λF αi +
X
F βz
z6=xi+1
where βz belongs to Homk(Z∆) (xi , z) and F z = F xi+1 . But F z is equal
to F xi+1 if and only if z is of the form τ r` xi+1 for an ` in Z. By the lemma,
we can write βz = βz0 αi . Thus we have the equality
X
(F ϕαi )Φxi = F (λIdxi+1 +
βz0 )F αi .
z
The scalar λ is not zero. Indeed, Φxi is an automorphism, thus the image of
(F ϕαi )Φxi is not zero in the quotient
R(F xi , F xi+1 )/R2 (F xi , F xi+1 ).
P
Thus Φxi+1 = F (λIdxi+1 + z βz0 ) is an automorphism of F xi+1 which
verifies the commutation relation
(F ϕαi ) ◦ Φxi = Φxi+1 ◦ F αi .
6.3. Other standard cases. — In the mesh category k(Z∆), where ∆ is a Dynkin
tree, the length of the non zero paths is bounded. Thus there exist automorphisms ϕ such that, for an arrow α : x → y of ∆, the paths from x to ϕr y vanish
in the mesh category for all r 6= 0. In other words, for each arrow α : x → y of
Z∆, we have
M
Homk(Z∆)/ϕZ (x, y) =
Homk(Z∆) (x, ϕr y) = Homk(Z∆) (x, y) ' k,
r∈Z
where k(Z∆)/ϕZ is the orbit category (see Section 6.1).
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Lemma 6.4. — Let T be a finite triangulated category with AR-quiver Γ =
Z∆/G. Let ϕ be a generator of G and suppose that ϕ verifies for each arrow x → y of Z∆
M
Homk(Z∆) (x, ϕr y) = Homk(Z∆) (x, y) ' k.
r∈Z
Let F : k(Z∆) → T and G : k(Z∆) → T be covering functors satisfying the
AR-property. Suppose that F and G agree up to isomorphism on the objects
of k(Z∆). Then F and G are isomorphic as k-linear functors.
Proof. — Using Lemma 5.3, we have just to construct an isomorphism between the functors restricted to ∆. Let α : x → y be an arrow of ∆. Using
Theorem 5.1 and the hypothesis, we have the isomorphisms
M
M
Homk(Z∆) (x, z) '
HomT (F x, F y) '
Homk(Z∆) (x, ϕr y) ' k
F z=F y
r∈Z
and then
HomT (Gx, Gy) ' HomT (F x, F y) ' k.
Thus there exists a scalar λ such that Gα = λF α. This scalar does not vanish
since F and G are covering functors. As ∆ is a tree, we can find some λx
for x ∈ ∆ by induction such that
Gα = λx λ−1
y F α.
Now it is easy to check that Φx = λx IdF x is the functor isomorphism.
This lemma gives us an isomorphism between the functors F and F ◦ ϕ,
Moreover, using the same argument, one can show that the covering functor F
is an S-functor and a τ -functor.
For each Dynkin tree ∆ we can determine the automorphisms ϕ which satisfy
this combinatorial property. Using the preceding lemma and the equivalence
criterion we deduce the following theorem:
Theorem 6.5. — Let T be a finite triangulated category with AR-quiver Γ =
Z∆/G. Let ϕ be a generator of G. If one of these cases holds,
• ∆ = An with n odd and G is generated by τ r or ϕ = τ r φ with r ≥ 21 (n − 1)
1
and φ = τ 2 (n+1) S;
• ∆ = An with n even and G is generated by ρr with r ≥ n − 1 and
1
ρ = τ 2 n S;
• ∆ = Dn with n ≥ 5 and G is generated by τ r or τ r φ with r ≥ n − 2 and
φ as in Theorem 2.1;
• ∆ = D4 and G is generated by φτ r , where r ≥ 2 and φ runs over σ3 ;
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• ∆ = E6 and G is generated by τ r or τ r φ where r ≥ 5 and φ is as in
Theorem 2.1;
• ∆ = E7 and G is generated by τ r , r ≥ 8;
• ∆ = E8 and G is generated by τ r , r ≥ 14;
then T is standard, i.e. the categories T and k(Γ) are equivalent as k-linear
categories.
Corollary 6.6. — A finite maximal d-Calabi-Yau (see [30, 8]) triangulated
category T , with d ≥ 2, is standard, i.e. there exists a k-linear equivalence
between T and the orbit category Db (mod k∆)/τ −1 S d−1 where ∆ is Dynkin of
type A, D or E
7. Algebraic case
For some automorphism groups G, we know the k-linear structure of T . But
what about the triangulated structure? We can only give an answer adding
hypothesis on the triangulated structure. We distinguish two cases.
If T is locally finite, not finite, we have the following theorem which is proved
in Section 7.2.
Theorem 7.1. — Let T be a connected locally finite triangulated category with
infinitely many indecomposables. If T is the base of a tower of triangulated
categories [29], then T is triangle equivalent to Db (mod k∆) for some Dynkin
diagram ∆.
Now if T is a finite standard category which is algebraic, i.e. T is triangle
equivalent to E for some k-linear Frobenius category E (see [31, 3.6]), then we
have the following result which is proved in Section 7.3.
Theorem 7.2. — Let T be a finite triangulated category, which is connected,
algebraic and standard. Then, there exists a Dynkin diagram ∆ of type A, D
or E and an auto-equivalence Φ of Db (mod k∆) such that T is triangle equivalent to the orbit category Db (mod k∆)/Φ.
This theorem combined with Corollary 6.6 yields the following result (compare to [30, Cor. 8.4]).
Corollary 7.3. — If T is a finite algebraic maximal d-Calabi-Yau category with d ≥ 2, then T is triangle equivalent to the orbit category
Db (mod k∆)/S d ν −1 for some Dynkin diagram ∆.
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7.1. ∂-functor. — We recall the following definition from [29] and [46].
Definition 7.1.1. — Let H be an exact category and T a triangulated category. A ∂-functor (I, ∂) : H → T is given by:
— an additive k-linear functor I : H → T ;
i
p
— for each conflation : X Y Z of H, a morphism ∂ : IZ → SIX
`Ip
Ii
∂
functorial in such that IX −→ IY −−→ IZ −→ SIX is a triangle of T .
For each exact category H, the inclusion I : H → Db (H) can be completed
to a ∂-functor (I, ∂) in a unique way. Let T and T 0 be triangulated categories.
If (F, ϕ) : T → T 0 is an S-functor and (I, ∂) : H → T is a ∂-functor, we say
that F respects ∂ if (F ◦ I, ϕ(F ∂)) : H → T 0 is a ∂-functor. Obviously each
triangle functor respects ∂.
Proposition 7.4. — Let H be a k-linear hereditary abelian category and let
(I, ∂) : H → T be a ∂-functor. Then there exists a unique (up to isomorphism)
k-linear S-functor F : Db (H) → T which respects ∂.
Proof. — On H (which can be seen as a full subcategory of Db (H)), the functor F is uniquely determined. We want F to be an S-functor, so F is uniquely
determined on S n H for n ∈ Z too. Since H is hereditary, each object of Db (H)
is isomorphic to a direct sum of stalk complexes, i.e. complexes concentrated
in a single degree. Thus, the functor F is uniquely determined on the objects.
Now, let X and Y be stalk complexes of Db (H) and f : X → Y a non-zero
morphism. We can suppose that X is in H and Y is in S n H. If n 6= 0, 1,
f is necessarily zero. If n = 0, then f is a morphism in H and F f is uniquely
determined. If n = 1, f is an element of Ext1H (X, S −1 Y ), so gives us a conflation
i
p
: S −1 Y E X in H. The functor F respects ∂, thus F f has to be equal
to ϕ ◦ ∂ where ϕ is the natural isomorphism between SF S −1 Y and F Y . Since
∂ is functorial, F is a functor. The result follows.
A priori this functor is not a triangle functor. We recall a theorem proved
by B. Keller [29, Cor. 2.7].
Theorem 7.5. — Let H be a k-linear exact category, and T be the base of a
tower of triangulated categories [29]. Let (I, ∂) : H → T be a ∂-functor such
that for each n < 0, and all objects X and Y of H, the space HomT (IX, S n IY )
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vanishes. Then there exists a triangle functor F : Db (H) → T such that the
following diagram commutes up to isomorphism of ∂-functors:
From Theorem 7.5, and the proposition above we deduce the following corollary.
Corollary 7.6 (compare [44]). — Let T , H and (I, ∂) : H → T be as in
Theorem 7.5. If H is hereditary, then the unique functor F : Db (H) → T
which respects ∂ is a triangle functor.
7.2. Proof of Theorem 7.1. — Let F be the k-linear equivalence constructed in
Theorem 5.1 between an algebraic triangulated category T and Db (H) where
H = mod k∆ and ∆ is a simply-laced Dynkin graph. As we saw in Section 6,
the covering functor is an S-functor.
The category H is the heart of the standard t-structure on Db (H). The
image of this t-structure through F is a t-structure on T . Indeed, F is an
S-equivalence, so the conditions (i) and (ii) from [7, Def. 1.3.1] hold obviously.
And since H is hereditary, for an object X of Db (H), the morphism τ>0 X →
Sτ≤0 X of the triangle
τ≤0 X −→ X −→ τ>0 X −→ Sτ≤0 X
vanishes. Thus the image of this triangle through F is a triangle of T and
condition (iii) of [7, Def. 1.3.1] holds. Then we get a t-structure on T whose
heart is H.
It results from [7, Prop. 1.2.4] that the inclusion of the heart of a t-structure
can be uniquely completed to a ∂-functor. Thus we obtain a ∂-functor (F0 , ∂) :
H → T with F0 = F H.
The functor F is an S-equivalence. Thus for each n < 0, and all objects
X and Y of H, the space HomT (F X, S n F Y ) vanishes. Now we can apply
Theorem 7.5 and we get the following commutative diagram
where F is the S-equivalence and G is a triangle functor. Note that a priori F is an S-functor which does not respect ∂. The functors F H and G H
are isomorphic. The functor F is an S-functor thus we have an isomorphism
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F S n H ' G S n H for each n ∈ Z. Thus the functor G is essentially surjective.
Since H is the category mod k∆, to show that G is fully faithful, we have just
to show that for each p ∈ Z, there is an isomorphism induced by G
∼
HomDb (H) (A, S p A) −
→ HomT (GA, S p GA)
where A is the free module k∆. For p = 0, this is clear because A is in H. And
for p 6= 0 both sides vanish.
Thus G is a triangle equivalence between Db (H) and T .
7.3. Finite algebraic standard case. — For a small dg category A, we denote by
CA the category of dg A-modules, by DA the derived category of A and by
per A the perfect derived category of A, i.e. the smallest triangulated subcategory of DA which is stable under passage to direct factors and contains the
free A-modules A(?, A), where A runs through the objects of A. Recall that
a small triangulated category is algebraic if it is triangle equivalent to per A
for a dg category A. For two small dg categories A and B, a triangle functor
per A → per B is algebraic if it is isomorphic to the functor
L
FX = ? ⊗A X
associated with a dg bimodule X, i.e. an object of the derived category
D(Aop ⊗ B).
Let Φ be an algebraic autoequivalence of Db (mod k∆) such that the orbit
category Db (mod k∆)/Φ is triangulated. Let Y be a dg k∆-k∆-bimodule such
that Φ = FY . In Section 9.3 of [30], it was shown that there is a canonical triangle equivalence between this orbit category and the perfect derived category
of a certain small dg category. Thus, the orbit category is algebraic, and endowed with a canonical triangle equivalence to the perfect derived category of
a small dg category. Moreover, by the construction in [loc. cit.], the projection
functor
π : Db (mod k∆) −→ Db (mod k∆)/Φ
is algebraic.
The proof of Theorem 7.0.5 is based on the following universal property
of the triangulated orbit category Db (mod k∆)/Φ. For the proof, we refer to
Section 9.3 of [30].
Proposition 7.7. — Let B be a small dg category and
L
FX = ? ⊗k∆ X : Db (mod k∆) −→ per B
an algebraic triangle functor given by a dg k∆-A-bimodule X. Suppose that
there is an isomorphism between Y ⊗L
k∆ X and X in the derived bimodule
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category D(k∆op ⊗ B). Then the functor FX factors, up to isomorphism of
triangle functors, through the projection
π : Db (mod k∆) −→ Db (mod k∆)/Φ.
Moreover, the induced triangle functor is algebraic.
Let us recall a lemma of Van den Bergh [33].
Lemma 7.8. — Let Q be a quiver without oriented cycles and A be a dg category. We denote by k(Q) the category of paths of Q and by Can : CA → DA
the canonical functor. Then we have the following properties:
a) Each functor F : k(Q) → DA lifts, up to isomorphism, to a functor
‹
F : k(Q) → CA which verifies the following property: For each vertex j of Q,
the induced morphism
M
‹i → F
‹j,
F
i
where i runs through the immediate predecessors of j, is a monomorphism which
splits as a morphism of graded A-modules.
b) Let F and G be functors from k(Q) to CA, and suppose that F satisfies
the property of a). Then any morphism of functors ϕ : Can ◦ F → Can ◦ G lifts
to a morphism ϕ
e : F → G.
Proof. — a) For each vertex i of Q, the object F i is isomorphic in DA to its
cofibrant resolution Xi . Thus for each arrow α : i → j, F induces a morphism
fα : Xi → Xj which can be lifted to CA since the Xi are cofibrant. Since Q has
no oriented cycle, it is easy to choose the fα such that the property is satisfied.
b) For each vertex i of Q, we may assume that F i is cofibrant. Then we can
lift ϕi : Can ◦ F i → Can ◦ Gi to ψi : F i → Gi. For each arrow α of Q, the
square
F
F i −−−α−→ F j




ψi y
y ψj
G
α
Gi −−−−
→ Gj
is commutative in DA. Thus the square
M
F
F i −−−α−→ F j

i


 ψj
y
(ψi ) y
M
Gi −−−−→ Gj
i
Gα
L
is commutative up to nullhomotopic morphism h :
i F i → Gj. Since the
L
morphism f : i F i → F j is split mono in the category of graded A-modules, h
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extends along f and we can modify Ψj so that the square becomes commutative
in CA. The quiver Q does not have oriented cycles, so we can construct ϕ
e by
induction.
Proof of Theorem 7.2. — The category T is small and algebraic, thus we may
assume that T = per A for some small dg category A. Let F : Db (mod k∆) →
T be the covering functor of Theorem 5.1. Let Φ be an auto-equivalence of
Db (mod k∆) such that the AR-quiver of the orbit category Db (mod k∆)/Φ is
isomorphic (as translation quiver) to the AR-quiver of T . We may assume
op
that Φ = − ⊗L
⊗ k∆). The orbit category
k∆ Y for an object Y of D(k∆
b
D (mod k∆)/Φ is algebraic, thus it is per B for some dg category B.
‹ from k(∆) to CA. This
The functor F k(∆) lifts by Lemma 7.8 to a functor F
‹(k∆) has a structure of dg k∆op ⊗ A-module. We
means that the object X = F
denote by X the image of this object in D(k∆op ⊗ A).
The functors F and − ⊗L
k∆ X become isomorphic when restricted to k(∆).
Moreover − ⊗L
X
satisfies
the AR-property since it is a triangulated functor.
k∆
Thus by Lemma 5.3, they are isomorphic as k-linear functors. So we have the
following diagram:
The category T is standard, thus there exists an isomorphism of k-linear functors
L
L
L
c : − ⊗k∆ X −→ − ⊗k∆ Y ⊗k∆ X.
The functor − ⊗L
k∆ X restricted to the category k(∆) satisfies the property
of a) of Lemma 7.8. Thus we can apply b) and lift c k(∆) to an isomorphism c̃
op
between X and Y ⊗L
⊗ A-modules.
k∆ X as dg-k∆
By the universal property of the orbit category, the bimodule X endowed
with the isomorphism c̃ yields a triangle functor from Db (mod k∆)/Φ to T
which comes from a bimodule Z in D(B op ⊗ A).
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The functor − ⊗L
k∆ Z is essentially surjective. Let us show that it is fully faithful. For M and N objects of Db (mod k∆) we have the commutative diagram
where D means Db (mod k∆). The two diagonal morphisms are isomorphisms,
thus so is the horizontal morphism. This proves that − ⊗L
k∆ Z is a triangle
equivalence between the orbit category Db (mod k∆)/Φ and T .
8. Triangulated structure on the category of projectives
Let k be an algebraically closed field and P a k-linear category with split
idempotents. The category mod P of contravariant finitely presented functors
from P to mod k is exact. As the idempotents split, the projectives of mod P
coincide with the representables. Thus the Yoneda functor gives a natural
equivalence between P and proj P. Assume besides that mod P has a structure
of Frobenius category. The stable category mod P is a triangulated category,
we write Σ for the suspension functor.
Let S be an auto-equivalence of P. It can be extended to an exact functor
from mod P to mod P and thus to a triangle functor of mod P. The aim of
this part is to find a necessary condition on the functor S such that the category (P, S) has a triangulated structure. Heller already showed [25, Thm. 16.4]
that if there exists an isomorphism of triangle functors between S and Σ3 , then
P has a pretriangulated structure. But he did not succeed in proving the octahedral axiom. We are going to impose a stronger condition on the functor S
and prove the following theorem.
Theorem 8.1. — Assume there exists an exact sequence of exact functors
from mod P to mod P:
0 → Id −→ X 0 −→ X 1 −→ X 2 −→ S → 0,
where the X i , i = 0, 1, 2, take values in proj P. Then the category P has a
structure of triangulated category with suspension functor S.
For an M in mod P, denote TM : X 0 M → X 1 M → X 2 M → SX 0 M a stanu
v
w
dard triangle. A triangle of P will be a sequence X : P −
→ Q −
→ R −
→ SP
which is isomorphic to a standard triangle TM for an M in mod P.
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8.1. S-complexes, Φ-S-complexes and standard triangles. — Let Acp(mod P) be
the category of acyclic complexes with projective components. It is a Frobenius
category whose projective-injectives are the contractible complexes, i.e. the
complexes homotopic to zero. The functor Z 0 : Acp(mod P) → mod P which
sends a complex
x−1
x0
x1
· · · −→ X −1 −−→ X 0 −→ X 1 −→ · · ·
to the kernel of x0 is an exact functor. It sends the projective-injectives to
projective-injectives and induces a triangle equivalence between Acp(mod P)
and mod P.
Definition 8.1.1. — An object of Acp(mod P) is called an S-complex if it is
S-periodic, i.e. if it has the following form:
u
v
w
Su
· · · −→ P −
→Q−
→R−
→ SP −−→ SQ −→ · · · .
The category S-comp of S-complexes with S-periodic morphisms is a non
full subcategory of Acp(mod P). It is a Frobenius category. The projectiveinjectives are the S-contractibles, i.e. the complexes homotopic to zero with
an S-periodic homotopy. Using the functor Z 0 , we get an exact functor from
S-comp to mod P which induces a triangle functor
Z 0 : S-comp −→ mod P.
Fix a sequence as in Theorem 8.1. Clearly, it induces for each object M of
mod P, a functorial isomorphism in mod P, ΦM : Σ3 M → SM.
Let Y be an S-complex:
u
v
w
Su
Y : · · · −→ P −
→Q−
→R−
→ SP −−→ SQ −→ · · ·
Let M be the kernel of u. Then Y induces an isomorphism θ (in mod P) between
Σ3 M and SM . If θ is equal to ΦM , we will say that X is a Φ-S-complex.
Let M be an object of mod P. The standard triangle TM can be see as a
Φ-S-complex:
· · · −→ X 0 M −→ X 1 M −→ X 2 M −→ SX 0 M −→ SX 1 M −→ · · ·
The functor T which sends an object M of mod P to the S-complex TM
is exact since the X i are exact. It satisfies the relation Z 0 ◦ T ' Idmod P .
Moreover, as it preserves the projective-injectives, it induces a triangle functor
T : mod P −→ S-comp.
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8.2. Properties of the functors Z 0 and T
Lemma 8.2. — An S-complex which is homotopy-equivalent to a Φ-S-complex
is a Φ-S-complex.
u
v
w
Proof. — Let X : P −
→Q−
→R−
→ SP be an S-complex homotopy-equivalent
0
v0
u
w0
to the Φ-S-complex X 0 : P 0 −→ Q0 −→ R0 −→ SP 0 . Let M be the kernel of u
and M 0 the kernel of u0 . By assumption, there exists a S-periodic homotopy
equivalence f from X to X 0 , which induces a morphism g = Z 0 f : M → M 0 .
Thus, we get the following commutative diagram:
The morphism g is an isomorphism of mod P since f is an isomorphism of
S-comp. Thus the morphisms Σ3 g and Sg are isomorphisms of mod P. The
following equality in mod P
θ = (Sg)−1 ΦM 0 Σ3 g = ΦM
shows that the complex X is a Φ-S-complex.
Lemma 8.3. — Let
u
v
w
X:P −
→Q−
→R−
→ SP,
u0
v0
w0
X 0 : P 0 −→ Q0 −→ R0 −→ SP 0
be two Φ-S-complexes. Suppose that we have a commutative square
u
P −−−−→


f0 y
Q

 1
yf
u
P 0 −−−−→ Q0 .
Then, there exists a morphism f 2 : R → R0 such that (f 0 , f 1 , f 2 ) extends to
an S-periodic morphism from X to X 0 .
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Proof. — Let M be the kernel of u, M 0 be the kernel of u0 and f : M → M 0 be
the morphism induced by the commutative square. As R and R0 are projectiveinjective objects, we can find a morphism g 2 : R → R0 such that the following
square commutes:
v
Q −−−−→

1 
f y
v0
R

 2
yg
Q0 −−−−→ R0 .
The morphism g 2 induces a morphism g : SM → SM 0 such that the following
square is commutative in mod P:
Φ
Σ3 M −−−M−→ SM




Σ3 f y
yg
Φ
0
M
Σ3 M 0 −−−
−→ SM 0 .
Thus the morphisms Sf and g are equal in mod P, i.e. there exists a projectiveinjective I of mod P and morphisms α : SM → I and β : I → SM 0 such
that g − Sf = βα. Let p (resp. p0 ) be the epimorphism from R onto SM
(resp. from R0 onto SM 0 ). Then, as I is projective, β factors through p0 .
We put f 2 = g 2 − γαp. Then obviously, we have the equalities f 2 v = v 0 f 1
and w0 f 2 = Sf 0 w. Thus the morphism (f 0 , f 1 , f 2 ) extends to a morphism
of S-comp.
Proposition 8.4. — The functor Z 0 : Φ-S-comp → mod P is full and essentially surjective. Its kernel is an ideal whose square vanishes.
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Proof. — The functor Z 0 is essentially surjective since we have the relation
Z 0 ◦ T = Idmod P . Let us show that Z 0 is full. Let
u
v
w
X:P −
→Q−
→R−
→ SP,
u0
v0
w0
X 0 : P 0 −→ Q0 −→ R0 −→ SP 0
be two Φ-S-complexes. Let M (resp. M 0 ) be the kernel of u (resp. u0 ). As P ,
Q, P 0 and Q0 are projective-injective, there exist morphisms f 0 : P → P 0 and
f 1 : Q → Q0 such that the following diagram commutes:
u
M
P −
→ Q
 




1 
 f f0 
f
y
y
y
u0
M 0 P 0 −→ Q0 .
Now the result follows from Lemma 8.3.
Now let f : X → X 0 be a morphism in the kernel of Z 0 . Up to homotopy,
we can suppose that f has the following form:
u
v
w
u0
v0
w0
P −−−−→ Q −−−−→ R −−−−→






0y
0y
f2 y
SP


0y
P 0 −−−−→ Q0 −−−−→ R0 −−−−→ SP 0 .
As the composition w0 f 2 vanishes and as Q0 is projective-injective, f 2 factors
through v 0 . For the same argument, f 2 factors through w. If f and f 0 are
composable morphisms of the kernel of Z 0 , we get the following diagram:
The composition f 0 ◦ f vanishes obviously.
Corollary 8.5. — A Φ-S-complex morphism f which induces an isomorphism Z 0 (f ) in mod P is an homotopy-equivalence.
This corollary comes from the previous theorem and from the following
lemma.
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Lemma 8.6. — Let F : C → C 0 be a full functor between two additive categories. If the kernel of F is an ideal whose square vanishes, then F detects
isomorphisms.
Proof. — Let u ∈ HomC (A, B) be a morphism in C such that F u is an isomorphism. Since the functor F is full, there exists v in HomC (B, A) such that
F v = (F u)−1 . The morphism w = uv − IdB is in the kernel of F , thus w2
vanishes. Then the morphism v(IdB − w) is a right inverse of u. In the same
way we show that u has a left inverse, so u is an isomorphism.
Proposition 8.7. — The category of Φ-S-complexes is equivalent to the category of S-complexes which are homotopy-equivalent to standard triangles.
Proof. — Since standard triangles are φ-S-complexes, each S-complex that is
homotopy equivalent to a standard triangle is a Φ-S-complex (Lemma 8.2).
u
v
w
Let X : P −
→Q−
→R−
→ SP be a Φ-S-complex. Let M be the kernel of u.
Then there exist morphisms f 1 : P → X 0 M and f 1 : Q → X 1 M such that the
following diagram is commutative:
M /
M /
u
/P
f0
/ X 0M
/Q
f1
/ X 1 M.
We can complete (Lemma 8.3) f into an S-periodic morphism from X in TM .
The morphism f satisfies Z 0 f = IdM , so Z 0 (TM ) and Z 0 (X) are equal
in mod P. By the corollary, TM and X are homotopy-equivalent. Thus the
inclusion functor T is essentially surjective.
These two diagrams summarize the results of this section:
Acp
Acp
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8.3. Proof of Theorem 8.1. — We are going to show that the Φ-S-complexes
form a system of triangles of the category P. We use triangle axioms as in [36].
TR0: For each object M of P, the S-complex M == M → 0 → SM is homotopy-equivalent to the zero complex, so is a Φ-S-complex.
TR1: Let u : P → Q be a morphism of P, and let M be its kernel. We can
find morphisms f 0 and f 1 so as to obtain a commutative square:
We form the push-out
0 → Coker a −−−−→ X 2 M −−−−→




γy
y
PO
SM → 0
0 → Coker u −−−−−→ R −−−−−→ SM → 0.
It induces a triangle morphism of the triangulated category mod P:
Coker a −−−−→ X 2 M −−−−→




γy
y
SM −−−−→ ΣCoker a


y Σγ
Coker u −−−−−→ R −−−−−→ SM −−−−→ ΣCoker u.
The morphism γ is an isomorphism in mod P since Coker a and Coker u are
canonically isomorphic to Σ2 M in mod P. By the five lemma, X 2 M → R is
an isomorphism in mod P. Since X 2 M is projective-injective, so is R. Thus
u
the complex P −
→ Q → R → SP is an S-complex. Then we have to see that it
is a Φ-S-complex. Let θ be the isomorphism between SM and Σ3 M induced
by this complex. We write α (resp. β) for the canonical isomorphism in mod P
between Σ2 M and Coker a (resp. Coker u). From the commutative diagram
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we deduce the equality θ = (Σβ)−1 ΣγΣαΦM = ΦM in mod P. The constructed
S-complex is a Φ-S-complex.
u
v
w
TR2: Let X : P −
→ Q −
→ R −
→ SP be a Φ-S-complex. It is homotopyequivalent to a standard triangle TM . Thus the S-complex
−v
−w
−Su
X 0 : Q −−→ R −−→ SP −−−→ SQ
is homotopy-equivalent to TM [1]. Since T is a triangle functor, the objects
TΣM and TM [1] are isomorphic in the stable category S − comp, i.e. they are
homotopy-equivalent. Thus, by Lemma 8.2, TM [1] is a Φ-S-complex and then
so is X 0 .
TR3: This axiom is a direct consequence of Lemma 8.3.
TR4: Let X and X 0 be two Φ-S-complexes and suppose we have a commutative diagram:
u
X : P −−−−→

 0
yf
u0
v
w
v0
w0
Q −−−−→ R −−−−→

 1
yf
SP

 0
y Sf
X 0 : P 0 −−−−→ Q0 −−−−→ R0 −−−−→ SP 0 .
Let M (resp. M 0 ) be the kernel of u (resp. u0 ), and g : M → M 0 the induced
morphism. The morphism T g : TM → TM 0 induces a S-complex morphism
ge = (g 0 , g 1 , g 2 ) between X and X 0 .
We are going to show that we can find a morphism f 2 : R → R0 such that
(f , f 1 , f 2 ) can be extended in an S-complex morphism that is homotopic to ge.
As (g 0 , g 1 ) and (f 0 , f 1 ) induce the same morphism g in the kernels, we have
some morphisms h1 : Q → P 0 and h2 : R → Q0 such that f 0 − g 0 = h1 u and
f 1 − g 1 = u0 h1 + h2 v. We put f 2 = g 2 + v 0 h2 . We have the equalities
0
f 2 v = g 2 v + v 0 h2 v = v 0 (g 1 + h2 v) = v 0 (f 1 − u0 h1 ) = v 0 f 1 ,
w0 f 2 = w0 g 2 = (Sg 0 )w = (Sf 0 − Sh1 Su)w = (Sf 0 )w.
Thus (f 0 , f 1 , f 2 ) can be extended to an S-periodic morphism fe which is Shomotopic to ge. Their respective cones C(fe) and C(e
g ) are isomorphic as
S-complexes. Moreover, since ge is a composition of T g : TM → TM 0 with
homotopy-equivalences, the cones C(e
g ) and C(T g) are homotopy-equivalent.
In mod P, we have a triangle
g
M−
→ M 0 −→ C(g) −→ ΣM.
Since T is a triangle functor, the sequence
Tg
TM −→ TM 0 −→ TC(g) −→ TΣM
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is a triangle in S-comp. But we know that
Tg
TM −−→ TM 0 −→ C(T g) −→ TM [1]
is a triangle in S-comp. Thus the objects C(T g) and TC(g) are isomorphic in S-comp, i.e. homotopy-equivalent. Thus, the cone C(fe) of fe is a
Φ-S-complex by Lemma 8.2.
9. Application to the deformed preprojective algebras
In this section, we apply Theorem 8.1 to show that the category of finite
dimensional projective modules over a deformed preprojective algebra of generalized Dynkin type (see [8]) is triangulated. This will give us some examples
of non standard triangulated categories with finitely many indecomposables.
9.1. Preprojective algebra of generalized Dynkin type. — Recall the notations of
[8]. Let ∆ be a generalized Dynkin graph of type An , Dn (n ≥ 4), En (n =
6, 7, 8), or Ln . Let Q∆ be the following associated quiver:
The preprojective algebra P (∆) associated to the graph ∆ is the quotient of
the path algebra kQ∆ by the relations
X
aa, for each vertex i of Q∆ .
sa=i
The following proposition is classical [8, Prop 2.1].
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Proposition 9.1. — The preprojective algebra P (∆) is finite dimensional
and selfinjective. Its Nakayama permutation ν is the identity for ∆ = A1 , D2n ,
E7 , E8 and Ln , and is of order 2 in all other cases.
9.2. Deformed preprojective algebras of generalized Dynkin type. — Let us recall
the definition of deformed preprojective algebra introduced by [8]. Let ∆ be
a graph of generalized Dynkin type. We define an associated algebra R(∆) as
follows:
R(An ) = k, R(Dn ) = khx, yi/ x2 , y 2 , (x + y)n−2 ,
R(En ) = khx, yi/ x2 , y 3 , (x + y)n−3 , R(Ln ) = k[x]/(x2n ).
Further, we fix an exceptional vertex in each graph as follows (with the notations of the previous section):
0 for ∆ = An or Ln ,
2 for ∆ = Dn ,
3 for ∆ = En .
2
Let f be an element of the square rad R(∆) of the radical of R(∆). The
deformed preprojective algebra P f (∆) is the quotient of the path algebra kQ∆
by the relations
X
aa, for each non exceptional vertex i of Q,
sa=i
and
a0 a0
for ∆ = An ,
a0 a0 + a1 a1 + a2 a2 + f (a0 a0 , a1 a1 ), and (a0 a0 + a1 a1 )n−2
for ∆ = Dn ,
a0 a0 + a2 a2 + a3 a3 + f (a0 a0 , a2 a2 ), and (a0 a0 + a2 a2 )n−3
for ∆ = En ,
2 + a0 a0 + f (), and 2n
for ∆ = Ln .
Note that if f is zero, we get the preprojective algebra P (∆).
9.3. Corollaries of [8]. — The following proposition [8, Prop. 3.4] shows that the
category proj P f (∆) of finite-dimensional projective modules over a deformed
preprojective algebra satisfies the hypothesis of Theorem 8.1.
Proposition 9.2. — Let A = P f (∆) be a deformed preprojective algebra.
Then there exists an exact sequence of A-A-bimodules
0 → 1 AΦ−1 −→ P2 −→ P1 −→ P0 −→ A → 0,
where Φ is an automorphism of A and where the Pi ’s are projective as bimodules. Moreover, for each idempotent ei of A, we have Φ(ei ) = eν(i) .
So we can easily deduce the corollary:
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Corollary 9.3. — Let P f (∆) be a deformed preprojective algebra of generalized Dynkin type. Then the category proj P f (∆) of finite dimensional projective
modules is triangulated. The suspension is the Nakayama functor.
Indeed, if Pi = ei A is a projective indecomposable, then Pi ⊗A AΦ is equal
to Φ(ei )A = eν(i) A thus to ν(Pi ).
Now we are able to answer to the question of the previous part and find a triangulated category with finitely many indecomposables which is not standard.
The proof of the following theorem comes essentially from [8], Theorem 1.3.
Theorem 9.4. — Let k be an algebraically closed field of characteristic 2.
Then there exist k-linear triangulated categories with finitely many indecomposables which are not standard.
Proof. — By [8, Thm. 1.3], we know that there exist basic deformed preprojective algebras of generalized Dynkin type P f (∆) which are not isomorphic
to P (∆). Thus the categories proj P f (∆) and proj P (∆) can not be equivalent. But both are triangulated by Corollary 9.3 and have the same AR-quiver
Z∆/τ = Q∆ .
Conversely, we have the following theorem:
Theorem 9.5. — Let T be a finite 1-Calabi-Yau triangulated category. Then
T is equivalent to proj Λ as k-category, where Λ is a deformed preprojective
algebra of generalized Dynkin type.
Proof. — Let M1 , . . . , Mn be representatives of the isoclasses of indecomposL
able objects of T . The k-algebra Λ = End( ni=1 Mi ) is basic, finite-dimensional
and selfinjective since T has a Serre duality. It is easy to see that T and proj Λ
are equivalent as k-categories.
Let mod Λ be the category of finitely presented Λ-modules. It is a Frobenius category. Denote by Σ the suspension functor of the triangulated category mod Λ. The category T is 1-Calabi-Yau, that is to say that the suspension
functor S of the triangulated category T and the Serre functor ν are isomorphic. But in mod Λ, the functors S and Σ3 are isomorphic. Thus, for each non
projective simple Λ-module M we have an isomorphism Σ3 M ' νM . We get
immediately the result by [8, Thm. 1.2].
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